A variety of techniques have been used in determining the presence and in estimating quantities of hydrocarbons (oil and gas) in earth formations. These methods are designed to determine parameters of interest, including among other things, porosity, fluid content, and permeability of the rock formation surrounding the wellbore drilled for recovering hydrocarbons. Typically, the tools designed to provide the desired information are used to log the wellbore. Much of the logging is done after the wellbores have been drilled. More recently, wellbores have been logged while drilling of the wellbores, which is referred to as measurement-while-drilling (“MWD”) or logging-while-drilling (“LWD”). Measurements have also been made when tripping a drillstring out of a wellbore: this is called measurement-while-tripping (“MWT”).
One evolving technique uses nuclear magnetic resonance (NMR) logging tools and methods for determining, among other things porosity, hydrocarbon saturation and permeability of the rock formations. NMR logging tools excite the nuclei of the fluids in the geological formations in the vicinity of the wellbore so that certain parameters such as spin density, longitudinal relaxation time (generally referred to in the art as “T1”), and transverse relaxation time (generally referred to as “T2”) of the geological formations can be estimated. From such measurements, formation parameters such as porosity, permeability and hydrocarbon saturation are determined, which provides valuable information about the make-up of the geological formations and the amount of extractable hydrocarbons.
A typical NMR tool generates a static magnetic field B0 in the vicinity of the wellbore and an oscillating field B1 in a direction perpendicular to B0. This oscillating field is usually applied in the form of short-duration pulses. The purpose of the B0 field is to polarize the magnetic moments of nuclei parallel to the static field and the purpose of the B1 field is to rotate the magnetic moments by an angle controlled by the width tp and the amplitude B1 of the oscillating pulse. For NMR logging, the most common sequence is the Carr-Purcell-Meiboom-Gill (“CPMG”) sequence that can be expressed asTW−90−(τ−180−τ−echo)n  (1)where TW is a wait time, 90 is a 90° tipping pulse, 180 and is a 180° refocusing pulse and 2τ=TE is the interecho spacing.
After being tipped by 90°, the magnetic moment precesses around the static field at a particular frequency known as the Larmor frequency ω, given by ω=γB0, where B0 is the field strength of the static magnetic field and γ is the gyromagnetic ratio. At the same time, the magnetic moments return to the equilibrium direction (i.e., aligned with the static field) according to a decay time known as the “spin-lattice relaxation time” or T1. Inhomogeneities of the B0 field result in dephasing of the magnetic moments and to remedy this, a 180° pulse is included in the sequence to refocus the magnetic moments. This refocusing gives a sequence of n echo signals. These echo sequences are then processed to provide information about the relaxation times.
Also associated with the spin of molecular nuclei is a second relaxation time, T2, called the transverse or spin—spin relaxation time. At the end of a 90° tipping pulse, all the spins are pointed in a common direction perpendicular, or transverse, to the static field, and they all precess at the Larmor frequency. However, because of small fluctuations in the static field induced by other spins or paramagnetic impurities, the spins precess at slightly different frequencies and the transverse magnetization dephases with a relaxation time T2.
Interpretation of NMR core or log data is often started by inverting the time-domain CPMG echo decay into a T2 parameter domain distribution. In general, the T2 of fluids in porous rocks depends on the pore-size distribution and the type and number of fluids saturating the pore system. Because of the heterogeneous nature of porous media, T2 decays exhibit a multiexponential behavior. The basic equation describing the transverse relaxation of magnetization in fluid saturated porous media is
                              M          ⁡                      (            t            )                          =                              ∫                          T                              2                ⁢                                                                  ⁢                min                                                    T                              2                ⁢                                                                  ⁢                max                                              ⁢                                    P              ⁡                              (                                  T                  2                                )                                      ⁢                          ⅇ                                                -                  t                                /                                  T                  2                                                      ⁢                          ⅆ                              T                2                                                                        (        2        )            where M is magnetization and effects of diffusion in the presence of a magnetic field gradient have not been taken into consideration. Eq.(2) is based on the assumption that diffusion effects may be ignored. In a gradient magnetic field, diffusion causes atoms to move from their original positions to new ones which also causes these atoms to acquire different phase shifts compared to atoms that did not move. This contributes to a faster rate of relaxation.
The effect of field gradients is given by an equation of the form
                              1                      T            2                          =                              1                          T                              2                ⁢                bulk                                              +                      1                          T                              2                ⁢                surface                                              +                      1                          T                              2                ⁢                diffusion                                                                        (        3        )            where the first two terms on the right hand side are related to bulk relaxation and surface relaxation while the third term is related to the field gradient G by an equation of the form
                              T                      2            ⁢            diffusion                          =                  C                                    TE              2                        ·                          G              2                        ·            D                                              (        4        )            where TE is the interecho spacing, C is a constant and D is the diffusivity of the fluid.
U.S. Pat. No. 6,512,371 to Prammer, et. al., discloses a well logging system and method for detecting the presence and estimating the quantity of gaseous and liquid hydrocarbons in the near-wellbore zone. The system uses a gradient-based, multiple-frequency NMR logging tool to extract signal components characteristic for each type of hydrocarbon. Measurements at different frequencies are interleaved to obtain, in a single logging pass, multiple data streams corresponding to different recovery times and/or diffusivity for the same spot in the formation.
One of the main difficulties in defining self-diffusion parameters of the fluid in the pore matrix is related to the fact that different fluids having the same relaxation times and different diffusion coefficients cannot be effectively separated. Due to the practical limitation of the signal-to-noise ratio, none of the existing inversion techniques allow an effective and stable reconstruction of both the relaxation and diffusion spectra.
Another difficulty in relaxation and diffusion spectra reconstruction is caused by internal magnetic gradients. Typically, the values of the internal gradients are unknown. Thus, the diffusion parameters cannot be correctly defined if the internal gradients are not considered in both the measurement and interpretation scheme.
To separate the relaxation and diffusion process, U.S. Pat. No. 6,597,171 to Hurlimann, et al. introduced the so-called diffusion editing sequence allowing an effective separation of the relaxation and diffusion process. The diffusion editing sequence is a short pulse sequence allowing different sensitivities of the measured NMR signal to the self-diffusion coefficients. Several different types of diffusion editing sequences are described in Hurlimann, '171, including an inversion recovery sequence and a driven equilibrium sequence. After the diffusion editing sequence a regular CPMG sequence with a short TE is applied to acquire NMR data. As demonstrated by Hurlimann '171 the diffusion editing technique allows for a 2D map with a relaxation time T2, in one direction and diffusion coefficient in a second direction. However, this technique fails to define the diffusion coefficient if internal magnetic field gradients are present in the formation.
The internal gradient issue, in relation to the diffusion measurements, was discussed in U.S. Pat. No. 5,698,979 to Taicher, et al., having the same assignee as the present application and the contents of which are incorporated herein by reference. Taicher suggested the use of dual frequency measurements when two different values of a static magnetic field within the porous medium are used to find decay rates for each frequency. It showed that under the assumption that internal gradients, Gi, are proportional to a static magnetic field and the difference in the magnetic susceptibility between the fluid and the solid matrix, the diffusion coefficients could be uniquely defined. The Taicher approach does not address differentiating fluids with the same relaxation times and different diffusion coefficients (in other words, generating a 2D map with a relaxation time, T2, in one direction and a diffusion coefficient in a second direction).
There is a need for a system and method of determining diffusion coefficients and relaxation times of fluids in an earth formation that takes into account internal field inhomogeneities. Such a system and method should preferably be efficient in terms of power consumption and acquisition time. The present invention satisfies this need and provides several advantages that would be recognized by those versed in the art.